Theorem rnun 5012

Description: Distributive law for range over union. Theorem 8 of [Suppes] p. 60. (Contributed by NM, 24-Mar-1998.)

Assertion

Ref	Expression
rnun	|- ran ( A u. B ) = ( ran A u. ran B )

Proof of Theorem rnun

Step	Hyp	Ref	Expression
1		cnvun 5009	. . . 4 |- `' ( A u. B ) = ( `' A u. `' B )
2	1	dmeqi 4805	. . 3 |- dom `' ( A u. B ) = dom ( `' A u. `' B )
3		dmun 4811	. . 3 |- dom ( `' A u. `' B ) = ( dom `' A u. dom `' B )
4	2, 3	eqtri 2186	. 2 |- dom `' ( A u. B ) = ( dom `' A u. dom `' B )
5		df-rn 4615	. 2 |- ran ( A u. B ) = dom `' ( A u. B )
6		df-rn 4615	. . 3 |- ran A = dom `' A
7		df-rn 4615	. . 3 |- ran B = dom `' B
8	6, 7	uneq12i 3274	. 2 |- ( ran A u. ran B ) = ( dom `' A u. dom `' B )
9	4, 5, 8	3eqtr4i 2196	1 |- ran ( A u. B ) = ( ran A u. ran B )

Syntax hints:   = wceq 1343   u. cun 3114   `'ccnv 4603   dom cdm 4604   ran crn 4605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-cnv 4612  df-dm 4614  df-rn 4615

This theorem is referenced by:  imaundi  5016  imaundir  5017  rnpropg  5083  fun  5360  foun  5451  fpr  5667  fprg  5668  sbthlemi6  6927  exmidfodomrlemim  7157